Enzymes III

Good, we got here so far. Last time we saw different types of reactions that go on in enzymes. The details of how this mechanisms work were determined by X-ray crystallography or NMR spectroscopy, by studying detailed 3D pictures of the active sites of enzymes with their substrated bound and unbound. However, many other types of studies have given us most of what we know about enzymes. The most modern ones include protein engeneering. We can change amino acids suspected to be crucial for the activity of the enzyme and see how detrimental (or favourable, although this is rare) the changes we did to the primary structure of the enzyme are.

However, X-ray, NMR, and PCR are things from the 70's, 80's, and 90's (not necessarilly in that order). So how the heck did people figured out what enzymes did before that? Most of the knowledge on enzyme mechanisms has been derived from the study of the rates of enzyme catalyzed reaction, and how these rates are affected by experimentally controlable parameters (substrate concentration, substrate chemical structure, pH, temperature, etc., etc.). These experiments study the kinetics of the enzyme reactions.

Background

It's usually good to start from the begining, so we will refresh a little kinetics in general. First we have to remember that even when we have a reaction whose stoichiometry looks like:

there can be a much larger number of elementary reactions (simple molecular processes) involved in the reaction:

I₁, I₂, ..., represent reaction intermediates, and their presence influences the kinetics of the reaction. We have to look at which of these elemental processes have effects on the rate of the reaction if we want to describe the kinetic behavior of the reaction.

For any of these simple molecular processes (reactions), the rate of the reaction will be directly proportional to the frequency at which the molecules involved come together (or hit each other productively), and therefore proportional to the concentrations involved in the reaction. For a simple reaction only involving the species A and P, with A G P, we have the the rate of the reaction, or the velocity, is equal to the change of [P] with time, which is also equal to the change of [A] with time, but with opposite sign:

In this case, we have presented a first order reaction: Our rate depends only on the first power of the concentration of our reactants. The order is the sum of the number of components our rate depends on. So, if our simple molecular process involves two molecules of a:

Our reaction rate will be:

the order of this reaction is now second order, because it depends on two molecules of A. Analogously, if our reaction is:

Our reaction rate will now depend both on A and B, abd it is also a second order reaction

Reactions that involve three molecules (third order) are very rate, because the chances of three molecules hitting each other at the same time are very very low. Now, how do we determine the rates of the reaction, or how the concentration of species change with time? Ugh! We have to use some math. since we have a differential equation, we need to integrate the rate equation as a function of time:

Remember that to integrate a differential equation we have to put everything that depends on [A] to one side, and everything that depends on t on the other. I we integrate between the initial concentration [A]_o and time zero, we get:

This is a decaying exponential. If we have a second order process depending only on [A] (that is, V = k x [A]²), integration gives us:

There is know a linear relation between 1/ [A] and t. In cases were we have a reaction of the type A + B G P, we usually keep one of the reactants at very high concentration, so that its concentration with time is almost constant, and we then consider the system as a pseudo-first order system. Using these type of equations, and measurements of concentrations and time we can calculate kinetic parameters.

Enzyme kinetics

So how does all this fits with enzymes? Well, the same principles apply, but they are a bit more complicated (why make it simple...). The reaction that we will first focus on is:

Although we have more proecess in a reaction like this, we can still break it down to simple molecular processes, and the overall rate of the reaction will always be dictated by the slower steps in it. The first researchers in the field found that for any particular enzyme, the rate of reaction would change with the initial concentration of substrate. However, after they increased the concentration of substrate a lot compared to the concentration of enzyme (S >> E), the rate did not depend on the substrate any more (it became zero order on [S]), but on the rate of release of product:

The first, non-rate limiting step, is the formation of the ES complex. This step is not rate limiting, and we see that there is an equilibrium established between ES and E + S. k₁ is the rate constant of the formation of the ES complex, usually known as k_on, and k_-1 is the release of substrate, usually called k_off. The second reaction involves the chemistry itself plus the release of the product from the enzyme, and is the rate-limiting step, with a rate constant k₂. For simplicity, we consider that this part of the reaction is irreversible, that it, it win't go back. In order to get the rate equation of this reaction, we have to look at the rate limiting step:

In the same way that we saw above, we can write the rate equation as:

Now, we do not know [ES], because we cannot measure it. The rate of formation of ES will depend both on the transformation of ES to products (k₂), as well as the binding (k₁) and unbinding (k_-1) of S from E:

We have to solve this differential for [ES], and this is pretty darn hard. In order to solve it, we have to make a number of assumptions, which are more or less OK:

- First, Michaelis and Menten proposed that if the first reactions (formation and destruction of the ES complex) are a lot faster than the catalytic step and the release of product, we can consider the the first part of the reaction, E + S G ES (the Michaelis complex), is in equilibrium, so:

K_s = k_-1 / k₁ = [E] [S] / [ES]

Now we could in principle integrate the equation laid out above, but we will still have an equation in which there are variables we cannot measure easily ([E], the ammount of free enzyme).

- We make a second assumption, and that is that the concentration of [ES] won't change with time for a period in the reaction. Does this make any sense? We first have to remember that the concentration of susbtrate is a lot higher than the concentration of enzyme. Right after the reaction starts and things stabilize, there will be a large excess of substrate. Thus, while [S] is large in relation to [E], the concentration of [ES] will remain almost constant, or in steady state:

In other words, as soon as some ES complex is broken down to products, some more will be formed at the same rate, because there is plenty S around, and the system is in equilibrium (th Michaelis Menten condition described above). This is therefore called the steady state assumption, and we are dealing with steady state kinetics. The moments before achieving steady state are called the pre-steady state. In the steady state, the ammount of [ES] complex that forms equals the ammount of [ES] complex that disapears, so:

d[ES] / dt = 0, or k₁[E] [S] = k_-1[ES] + k₂ [ES]

- Finally, we need to be able to meassure stuff to be able to get numbers. We cannot measure [ES] or [E] easily, but we can readily meassure (we actually know it from the begining) the total enzyme concentration, [E]_T:

[E]_T = [ES] + [E]

Substitution into the steady state relationship and rearrangement gives:

We make a new definition, the Michaelis constant K_M, defined as:

So now we can rearrange the equation we had above to:

And solving for [ES] we get:

Now we can go back to our rate equation, which was V = k₂ [ES], and we get:

This equation does not apply for all times, but only to the steady state, and it is therefore the initial velocity, or V_o. This is because after a certain time several things happen. First, the concentration of substrate will start to drop, and we will move away from the steady state. Second, as the reaction goes, we start getting more and more product, and the condition that the second reaction was irreversible may not apply: We may have product going back in order to establish its own thermodynamic equilibrium. With this in mind, we get:

The maximal initial velocity of the reaction occurs when [S] >> K_M, or we have the enzyme in saturation regime. At this point,

So we can recombine everything and get what is known as the Michaelis-Menten equation, which is the basic equation used to study enzyme kinetics:

As was the case with the binding of O₂ to myoglobin, this equation gives an hyperbolic plot:

Working also by analogy with myoglobin, K_M can be defined as the concentration of substrate at which the initial velocity is half of the maximal velocity. Therefore, if an enzyme has a small K_M it will achieve maximal catalytic efficiency at very low substrate concentration.

The Michaelis constant K_M is not a parameter of the enzyme alone, but of every enzyme-substrate pair, and it will be affected by the type of enzyme, type of substrate, temperature, and pH. If we remember the first condition that we stated, that the system was in equilibrium, and K_S = k_-1 / k₁, we can rewrite K_M as:

Since K_S is the dissociation constant of the ES complex, as K_S decreases, K_M will increase. Therefore, K_M is also a measure of the affinity of the enzyme for the substrate. This applies if k₂ < k_-1, and since we know k₂ is the rate limiting step, this will be fullfilled.

Before we finish this, we have yet another definition, the catalytic constant k_cat. It is defined as:

This number represent the number of reaction processes that a certain ammount of enzyme can do in a certain time, also known as the turnover number. For a simple system following Michaelis-Menten kinetics, we can see that k_cat = k₂. This makes perfect sense, because the turnover (the number of times the enzyme will catalyze the reaction per unit time) depends exclusivelly on the rate limiting step, which is determined by k₂.

Lineweaver-Burke (or double-reciprocal) plots

So far, so good. We have our rate for the initial velocity expressed with parameters that we can measure readily. Now, in order to get the kinetic parameters for any Michaelis-Menten system we need to obtain V_max and K_M (basically, the [S] at which V_o is V_max / 2).  To calculate V_max we could just increase the [S], because we know that the hyperbolic curve approaches asymptotically to V_max. Hoewever, this is hard to do in practice.

A much better method was proposed by Lineweaver and Burke in the 1930's. They took the reciprocal (that is 1 divided by the function) of the Michaelis-Menten equation. This is called the Lineweaver-Burke or double-reciprocal equation, and looks like this:

Instead of plotting V_o vs [S], we plot 1 / V_o vs 1 / [S], and we get a double-reciprocal plot:

As seen from this equation, there will be a linear relationship between 1 / V_o vs 1 / [S], and we can obtain the values of V_max and K_M from the Y and X intercepts, respectivelly. The Y intercept is 1 / V_max, and the X intercept (which is extrapolated, because 1 / [S] cannot be negative!) is -1 / K_M. One thing that we have to realize is that we will only get points for 1 / [S] > 0, and the error will grow as we decrease [S] (the value 1/ [S] will grow, but so will the error).

Not only are the Lineweaver-Burke plots useful to calculate kinetic parameters for an enzyme-substrate pair, but they are invaluable when studying enzymes that react with more than one substrate, and as we'll see later, enzyme inhibition.

Prepared by Guillermo Moyna, 1999.